2009 Linear Algebra Test B

江西财经大学

2009－2010学年第一学期期末考试试卷

试卷代码：12063B 授课课时：48

课程名称：Linear Algebra 适用对象：2008级国际学院

1. Filling in t he Blanks (3’×6=18’)

(1) If ????

?

?????=400270651A , then det(adj(A))= .

(2) If ?????

????

???=210011000053

0021A , then the inverse 1-A = (3)If ?????

????

???=00

0000

00b

a

a b b a a b A , then det(A 2)= (4) Let A be (3×3) matrix, and 3,1,6 are the eigenvalues of A . Then the eigenvalues of adj (A ) are .

(5) Let 162,713αβ????????=-=-????????????

. Then the tripe products )(βαα??= . (6) If the rank of ????

??????--=2111111

a a A r(A)=2, then parameter a = .

2. There are four choices in each question, but only one is correct. You should choose the correct one into the blank. (3’×6=18’)

(1) Let A and B be (n ×n) inverse matrices, then ( ) is not always correct. (A) B A B A +=+ (B) 111)(---=A B AB (C) )()()(A adj B adj AB adj = (D) B A AB =

(2) If A is an orthogonal matrix, and 1-=A , then adj(A)= . (A)T A (B) T A - (C)A (D) A -

(3) If the vectors set {}

p ααα,...,,21 is linear dependent, then the statement ( ) is always true.

(A)021=p ααα

(B) There is at least one non-zero vector among {}

p ααα,...,,21. (C)There are at least two proportional vectors among {}

p ααα,...,,21.

(D) There is at least one vector which is the linear combination of the rest vectors.

(4) Let A be (m n ?) matrix ，and n m <, then the statement ( ) is always true. (A) b AX = has infinitely solutions or has no solutions. (B) b AX = has infinitely solutions. (C) b AX = has no solutions.

(D) The above statements are all incorrect.

(5) If n n ? matrices A and B are similar, then the statement ( ) is always true. (A) A,B have the same eigenvalues and eigenvectors. (B) A,B only have the same eigenvectors. (C) The rank of matrices A,B, such that r(A)=r(B)

(D) The column vectors of A and B are all linear independent. (6) If A,B,P are all (n ×n) matrices, then ( )

(A)???

???=??????PB PA B A P (B) [][]PB PA B A P =

(C) 1111)(----=A B P ABP (D) T T T T P B A ABP =)(

3. (12’) If the system of linear equations is ???

??=++=-=+b

x ax x x x x x 321

31

2

111, then what values of b a , will make the system has only solution, infinitely many solutions, no solutions, and when the system has infinitely many solutions, find its all solutions.

4. (12’) Let 33)(?=ij a A matrix with 1,2,1131211-===a and a a Let ??

??

??????----=57923445

7C be the matrix of cofactors of A. (That is A 11=-7,A 12=5, and so on.) Find A.

5. (10’) Suppose A is a (2×2) matrix such that 032=-+I A A , and ??

?

???=??????=13,51u where Au .

Find u A and u A 32.

6. (15’) Find the all eigenvalues and all eigenvectors of matrix A, where ????

?

?????--=201034011A .

7. (10’) Determine whether the vectors set },,{321ααα, where

????

?

?

??????-=????????????-=????????????-=4323,3134,5312121ααα, is linear dependent or linear independent.

8. (5’) Suppose A is (n ×n ) orthogonal matrix, and |A|=1. Show that ：1=λ is an eigenvalue of A .